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- Advances in Mathematics of Communications
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DCDS

We introduce a notion of viscosity solutions for
a nonlinear degenerate diffusion equation with a drift potential. We
show that our notion of solutions coincide with the weak solutions
defined via integration by parts. As an application of the viscosity
solutions theory, we show that the free boundary uniformly converges
to the equilibrium as $t$ grows. In the case of a convex potential,
an exponential rate of free boundary convergence is obtained.

CPAA

We study the 1D contracting Stefan problem with two moving boundaries that describes the

*freezing*of a supercooled liquid. The problem is borderline ill--posed with a density in excess of unity indicative of the dividing line. We show that if the initial density, $\rho_0(x)$ does not exceed one and is not too close to one in the vicinity of the boundaries, then there is a unique solution for all times which is smooth for all positive times. Conversely if the initial density is too large, singularities may occur. Here the situation is more complex: the solution may suddenly freeze without any hope of continuation or may continue to evolve after a local instant freezing but, sometimes, with the loss of uniqueness.
DCDS

The earlier paper [2] contains a lower bound of the solution in terms of its $L^1$ norm, which is incorrect. In this note we explain the mistake and present a correction to it under the restriction that the permeability constant $m$ satisfies $1< m <2$. As a consequence, the quantitative estimates on the convergence rate (Main Theorem (c) and Theorem 3.6 in [2] ) only hold for $1<\m<2$. For $\m\geq 2$ a partial convergence rate is obtained.

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